MATH 311–505/506
Fall 2009
Sample problems for the final exam
Any problem may be altered or replaced by a different one!
Problem 1 (15 pts.)
Find a quadratic polynomial p(x) = ax2 + bx + c such that
p(−1) = p(3) = 6 and p′ (2) = p(1).
Problem 2 (20 pts.) Let v1 = (1, 1, 1), v2 = (1, 1, 0), and v3 = (1, 0, 1). Let L : R3 → R3
be a linear operator on R3 such that L(v1 ) = v2 , L(v2 ) = v3 , L(v3 ) = v1 .
(i) Show that the vectors v1 , v2 , v3 form a basis for R3 .
(ii) Find the matrix of the operator L relative to the basis v1 , v2 , v3 .
(iii) Find the matrix of the operator L relative to the standard basis.

1 1 1
Problem 3 (20 pts.) Let B =  1 1 1 .
1 1 1

(i) Find all eigenvalues of the matrix B.
(ii) Find a basis for R3 consisting of eigenvectors of B?
(iii) Find an orthonormal basis for R3 consisting of eigenvectors of B?
Problem 4 (20 pts.) Find a quadratic polynomial q that is the best least squares fit to
the function f (x) = |x| on the interval [−1, 1]. This means that q should minimize the distance
Z 1
1/2
2
dist(f, q) =
|f (x) − q(x)| dx
−1
over all polynomials of degree at most 2.
Problem 5 (25 pts.) It is known that
2
Z
x
2x
2
2
x sin(ax) dx = − + 3 cos(ax) + 2 sin(ax) + C,
a
a
a
a 6= 0.
(i) Find the Fourier sine series of the function f (x) = x2 on the interval [0, π].
(ii) Over the interval [−3.5π, 3.5π], sketch the function to which the series converges.
(iii) Describe how the answer to (ii) would change if we studied the Fourier cosine series
instead.
Bonus Problem 6 (15 pts.)
equation
∂u
∂2u
=
∂t
∂x2
Solve the initial-boundary value problem for the heat
(0 < x < π,
t > 0),
u(x, 0) = 1 + 2 cos(2x) − cos(3x)
∂u
∂u
(0, t) =
(π, t) = 0
∂x
∂x
(t > 0).
(0 < x < π),